In the long term development of the research on wind waves and their modelling, the present situation is framed with a short look at the past, a critical analysis of the present capabilities and a foresight of where the field is likely to go. After a short introduction, Chapter 2 deals with the basic processes at work and their modelling aspects. Chapter 3 highlights the interaction with wind and currents. Chapter 4 stresses the need for a more complete, spectral, approach in data assimilation. Chapter 5 summarizes the situation with a discussion on the present status in wave modelling and a look at what we can expect in the future.

The dynamics of internal gravity waves is modelled using Wentzel–Kramer–Brillouin (WKB) theory in position–wave number phase space. A transport equation for the phase‐space wave‐action density is derived for describing one‐dimensional wave fields in a background with height‐dependent stratification and height‐ and time‐dependent horizontal‐mean horizontal wind, where the mean wind is coupled to the waves through the divergence of the mean vertical flux of horizontal momentum associated with the waves. The phase‐space approach bypasses the caustics problem that occurs in WKB ray‐tracing models when the wave number becomes a multivalued function of position, such as in the case of a wave packet encountering a reflecting jet or in the presence of a time‐dependent background flow. Two numerical models were developed to solve the coupled equations for the wave‐action density and horizontal mean wind: an Eulerian model using a finite‐volume method and a Lagrangian ‘phase‐space ray tracer’ that transports wave‐action density along phase‐space paths determined by the classical WKB ray equations for position and wave number. The models are used to simulate the upward propagation of a Gaussian wave packet through a variable stratification, a wind jet and the mean flow induced by the waves. Results from the WKB models are in good agreement with simulations using a weakly nonlinear wave‐resolving model, as well as with a fully nonlinear large‐eddy‐simulation model. The work is a step toward more realistic parametrizations of atmospheric gravity waves in weather and climate models.

Nonlinear interactions between sea waves and the sea bottom are a major mechanism for energy transfer between the different wave frequencies in the near-shore region. Nevertheless, it is difficult to account for this phenomenon in stochastic wave forecasting models due to its mathematical complexity, which mostly consists of computing either the bispectral evolution or non-local shoaling coefficients. In this work, quasi-two-dimensional stochastic energy evolution equations are derived for dispersive water waves up to quadratic nonlinearity. The bispectral evolution equations are formulated using stochastic closure. They are solved analytically and substituted into the energy evolution equations to construct a stochastic model with non-local shoaling coefficients, which includes nonlinear dissipative effects and slow time evolution. The nonlinear shoaling mechanism is investigated and shown to present two different behaviour types. The first consists of a rapidly oscillating behaviour transferring energy back and forth between wave harmonics in deep water.Owing to the contribution of bottom components for closing the class III Bragg resonance conditions, this behaviour includes mean energy transfer when waves reach intermediate water depths. The second behaviour relates to one-dimensional shoaling effects in shallow water depths. In contrast to the behaviour in intermediate water depths, it is shown that the nonlinear shoaling coefficients refrain from their oscillatory nature while presenting an exponential energy transfer. This is explained through the one-dimensional satisfaction of the Bragg resonance conditions by wave triads due to the non-dispersive propagation in this region even without depth changes. The energy evolution model is localized using a matching approach to account for both these behaviour types. The model is evaluated with respect to deterministic ensembles, field measurements and laboratory experiments while performing well in modelling monochromatic superharmonic self-interactions and infra-gravity wave generation from bichromatic waves and a realistic wave spectrum evolution. This lays physical and mathematical grounds for the validity of unexplained simplifications in former works and the capability to construct a formulation that consistently accounts for nonlinear energy transfers from deep to shallow water.

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